src/HOL/Univ.thy
author paulson
Thu Sep 23 13:06:31 1999 +0200 (1999-09-23)
changeset 7584 5be4bb8e4e3f
parent 7255 853bdbe9973d
child 8735 bb2250ac9557
permissions -rw-r--r--
tidied; added lemma restrict_to_left
     1 (*  Title:      HOL/Univ.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Declares the type ('a, 'b) node, a subtype of (nat=>'b+nat) * ('a+nat)
     7 
     8 Defines "Cartesian Product" and "Disjoint Sum" as set operations.
     9 Could <*> be generalized to a general summation (Sigma)?
    10 *)
    11 
    12 Univ = Arith + Sum +
    13 
    14 setup arith_setup
    15 
    16 
    17 (** lists, trees will be sets of nodes **)
    18 
    19 global
    20 
    21 typedef (Node)
    22   ('a, 'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
    23 
    24 types
    25   'a item = ('a, unit) node set
    26   ('a, 'b) dtree = ('a, 'b) node set
    27 
    28 consts
    29   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
    30   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
    31 
    32   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
    33   ndepth    :: ('a, 'b) node => nat
    34 
    35   Atom      :: "('a + nat) => ('a, 'b) dtree"
    36   Leaf      :: 'a => ('a, 'b) dtree
    37   Numb      :: nat => ('a, 'b) dtree
    38   Scons     :: [('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree
    39   In0,In1   :: ('a, 'b) dtree => ('a, 'b) dtree
    40 
    41   Lim       :: ('b => ('a, 'b) dtree) => ('a, 'b) dtree
    42   Funs      :: "'u set => ('t => 'u) set"
    43 
    44   ntrunc    :: [nat, ('a, 'b) dtree] => ('a, 'b) dtree
    45 
    46   uprod     :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
    47   usum      :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
    48 
    49   Split     :: [[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
    50   Case      :: [[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
    51 
    52   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
    53                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    54   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
    55                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    56 
    57 
    58 local
    59 
    60 defs
    61 
    62   Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    63 
    64   (*crude "lists" of nats -- needed for the constructions*)
    65   apfst_def  "apfst == (%f (x,y). (f(x),y))"
    66   Push_def   "Push == (%b h. nat_case b h)"
    67 
    68   (** operations on S-expressions -- sets of nodes **)
    69 
    70   (*S-expression constructors*)
    71   Atom_def   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
    72   Scons_def  "Scons M N == (Push_Node (Inr 1) `` M) Un (Push_Node (Inr 2) `` N)"
    73 
    74   (*Leaf nodes, with arbitrary or nat labels*)
    75   Leaf_def   "Leaf == Atom o Inl"
    76   Numb_def   "Numb == Atom o Inr"
    77 
    78   (*Injections of the "disjoint sum"*)
    79   In0_def    "In0(M) == Scons (Numb 0) M"
    80   In1_def    "In1(M) == Scons (Numb 1) M"
    81 
    82   (*Function spaces*)
    83   Lim_def "Lim f == Union {z. ? x. z = Push_Node (Inl x) `` (f x)}"
    84   Funs_def "Funs S == {f. range f <= S}"
    85 
    86   (*the set of nodes with depth less than k*)
    87   ndepth_def "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
    88   ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
    89 
    90   (*products and sums for the "universe"*)
    91   uprod_def  "uprod A B == UN x:A. UN y:B. { Scons x y }"
    92   usum_def   "usum A B == In0``A Un In1``B"
    93 
    94   (*the corresponding eliminators*)
    95   Split_def  "Split c M == @u. ? x y. M = Scons x y & u = c x y"
    96 
    97   Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) 
    98                                | (? y . M = In1(y) & u = d(y))"
    99 
   100 
   101   (** equality for the "universe" **)
   102 
   103   dprod_def  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
   104 
   105   dsum_def   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un 
   106                           (UN (y,y'):s. {(In1(y),In1(y'))})"
   107 
   108 end